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A vector space is a collection of objects, which we call vectors, that can be added together and multiplied by scalars, which are numbers. To be a vector space, certain rules must be followed. For instance, the sum of any two vectors must also be a vector in the space, and scalar multiplication must behave predictably. In the case we're discussing, our vector space is \( \mathscr{P}_{2} \), which consists of all polynomials that have a degree of at most 2.
This means that any polynomial in this space can be written as \( ax^2 + bx + c \), where the coefficients \( a \), \( b \), and \( c \) are real numbers. These polynomials can represent vectors.
Understanding vector spaces provides a framework for studying polynomials using linear algebra, where operations are well-defined and structured.

Linear independence is a key concept in linear algebra. A set of vectors is considered linearly independent if no vector in the set is a linear combination of the others. This means you cannot write one vector as a sum of the multiples of the others in the set.
In the case of our set \( \mathcal{B} = \{x, 1-x, 1+x+x^{2}\} \), to check for linear independence, we see if the equation

• \( a(x) + b(1-x) + c(1+x+x^{2}) = 0 \)

only holds true when \( a = b = c = 0 \).
If this system of coefficients leads to a trivial solution, where all coefficients are zero, the set is linearly independent.
In our example, solving the mathematical formulation resulted in \( a = b = c = 0 \), confirming that \( \mathcal{B} \) is indeed linearly independent.

A basis of a vector space is a set of vectors that not only spans the space but is also linearly independent. This set allows us to express every vector in the space uniquely as a combination of basis vectors. For a vector space with polynomials of degree up to 2, such as \( \mathscr{P}_{2} \), a basis has exactly three vectors.
In simple terms:

• Spanning means any vector in the space can be formed by a combination of vectors in the set.
• To confirm a set is a basis, check both linear independence and the ability to span the entire space.

In this scenario, the set \( \mathcal{B} = \{x, 1-x, 1+x+x^{2}\} \) is both linearly independent and spans \( \mathscr{P}_{2} \), making it a basis.

Polynomials are algebraic expressions that involve sums of powers of a variable. They appear frequently in mathematics and are a fundamental part of linear algebra when dealing with polynomial vector spaces.
In our exercise, the vector space \( \mathscr{P}_{2} \) is made up of polynomials with a degree of 2 or less, meaning they can be expressed as \( ax^2 + bx + c \). These polynomials are treated similarly to coordinate vectors but in a more abstract form.
Understanding how to work with polynomials in a vector space involves grasping their structure and how they can be manipulated using operations like addition and scalar multiplication.
By mastering these concepts, one gains a powerful toolset to tackle more complex algebraic problems.